Rigidity of minimal hypersurfaces of spheres with two principal curvatures

Abstract

Let M be a compact oriented minimal hypersurface of the unit n-dimensional sphere S n . It is known that if the norm squared of the second fundamental form, $$ ||II||^{2} : M \to {\bf R} $$ , satisfies that $$ ||II||^{2}(m) = n - 1 $$ for all $$ m \in M $$ , then M is isometric to a Clifford minimal hypersurface ([2], [5]). In this paper we will generalize this result for minimal hypersurfaces with two principal curvatures and dimension greater than 2. For these hypersurfaces we will show that if the average of the function $$ ||II||^{2} $$ is n - 1, then M must be a Clifford hypersurface.